PHYSICAL REVIEW D 103, 104047 (2021)

Constraints on black-hole charges with the 2017 EHT observations of M87*

– Prashant Kocherlakota ,1 Luciano Rezzolla,1 3 Heino Falcke,4 Christian M. Fromm,5,6,1 Michael Kramer,7 Yosuke Mizuno,8,9 Antonios Nathanail,9,10 H´ector Olivares,4 Ziri Younsi,11,9 Kazunori Akiyama,12,13,5 Antxon Alberdi,14 Walter Alef,7 Juan Carlos Algaba,15 Richard Anantua,5,6,16 Keiichi Asada,17 Rebecca Azulay,18,19,7 Anne-Kathrin Baczko,7 David Ball,20 Mislav Baloković,5,6 John Barrett,12 Bradford A. Benson,21,22 Dan Bintley,23 Lindy Blackburn,5,6 Raymond Blundell,6 Wilfred Boland,24 Katherine L. Bouman,5,6,25 Geoffrey C. Bower,26 Hope Boyce,27,28 – Michael Bremer,29 Christiaan D. Brinkerink,4 Roger Brissenden,5,6 Silke Britzen,7 Avery E. Broderick,30 32 Dominique Broguiere,29 Thomas Bronzwaer,4 Do-Young Byun,33,34 John E. Carlstrom,35,22,36,37 Andrew Chael,38,39 Chi-kwan Chan,20,40 Shami Chatterjee,41 Koushik Chatterjee,42 Ming-Tang Chen,26 Yongjun Chen (陈永军),43,44 Paul M. Chesler,5 Ilje Cho,33,34 Pierre Christian,45 John E. Conway,46 James M. Cordes,41 Thomas M. Crawford,22,35 Geoffrey B. Crew,12 Alejandro Cruz-Osorio,9 Yuzhu Cui,47,48 Jordy Davelaar,49,16,4 Mariafelicia De Laurentis,50,9,51 – Roger Deane,52 54 Jessica Dempsey,23 Gregory Desvignes,55 Sheperd S. Doeleman,5,6 Ralph P. Eatough,56,7 Joseph Farah,6,5,57 Vincent L. Fish,12 Ed Fomalont,58 Raquel Fraga-Encinas,4 Per Friberg,23 H. Alyson Ford,59 Antonio Fuentes,14 Peter Galison,5,60,61 Charles F. Gammie,62,63 Roberto García,29 Olivier Gentaz,29 Boris Georgiev,31,32 Ciriaco Goddi,4,64 Roman Gold,65,30 Jos´e L. Gómez,14 Arturo I. Gómez-Ruiz,66,67 Minfeng Gu (顾敏峰),43,68 Mark Gurwell,6 Kazuhiro Hada,47,48 Daryl Haggard,27,28 Michael H. Hecht,12 Ronald Hesper,69 Luis C. Ho (何子山),70,71 – Paul Ho,17 Mareki Honma,47,48,72 Chih-Wei L. Huang,17 Lei Huang (黄磊),43,68 David H. Hughes,66 Shiro Ikeda,13,73 75 Makoto Inoue,17 Sara Issaoun,4 David J. James,5,6 Buell T. Jannuzi,20 Michael Janssen,7 Britton Jeter,31,32 Wu Jiang (江悟),43 Alejandra Jimenez-Rosales,4 Michael D. Johnson,5,6 Svetlana Jorstad,76,77 Taehyun Jung,33,34 Mansour Karami,30,31 Ramesh Karuppusamy,7 Tomohisa Kawashima,78 Garrett K. Keating,6 Mark Kettenis,79 Dong-Jin Kim,7 Jae-Young Kim,33,7 Jongsoo Kim,33 Junhan Kim,20,25 Motoki Kino,13,80 Jun Yi Koay,17 Yutaro Kofuji,47,72 Patrick M. Koch,17 Shoko Koyama,17 Carsten Kramer,29 Thomas P. Krichbaum,7 Cheng-Yu Kuo,81,17 Tod R. Lauer,82 Sang-Sung Lee,33 Aviad Levis,25 Yan-Rong Li (李彦荣),83 Zhiyuan Li (李志远),84,85 Michael Lindqvist,46 Rocco Lico,14,7 Greg Lindahl,6 Jun Liu (刘俊),7 Kuo Liu,7 Elisabetta Liuzzo,86 Wen-Ping Lo,17,87 Andrei P. Lobanov,7 Laurent Loinard,88,89 – Colin Lonsdale,12 Ru-Sen Lu (路如森),43,44,7 Nicholas R. MacDonald,7 Jirong Mao (毛基荣),90 92 Nicola Marchili,86,7 Sera Markoff,42,93 Daniel P. Marrone,20 Alan P. Marscher,76 Iván Martí-Vidal,18,19 Satoki Matsushita,17 Lynn D. Matthews,12 Lia Medeiros,94,20 Karl M. Menten,7 Izumi Mizuno,23 James M. Moran,5,6 Kotaro Moriyama,12,47 Monika Moscibrodzka,4 Cornelia Müller,7,4 Gibwa Musoke,42,4 Alejandro Mus Mejías,18,19 Hiroshi Nagai,13,48 Neil M. Nagar,95 Masanori Nakamura,96,17 Ramesh Narayan,5,6 Gopal Narayanan,97 Iniyan Natarajan,54,52,98 Joseph Neilsen,99 Roberto Neri,29 Chunchong Ni,31,32 Aristeidis Noutsos,7 Michael A. Nowak,100 Hiroki Okino,47,72 Gisela N. Ortiz-León,7 Tomoaki Oyama,47 Feryal Özel,20 Daniel C. M. Palumbo,5,6 Jongho Park,17 Nimesh Patel,6 – Ue-Li Pen,30,101 103 Dominic W. Pesce,5,6 Vincent Pi´etu,29 Richard Plambeck,104 Aleksandar PopStefanija,97 Oliver Porth,42,9 Felix M. Pötzl,7 Ben Prather,62 Jorge A. Preciado-López,30 Dimitrios Psaltis,20 Hung-Yi Pu,105,17,30 Venkatessh Ramakrishnan,95 Ramprasad Rao,26 Mark G. Rawlings,23 Alexander W. Raymond,5,6 Angelo Ricarte,5,6 Bart Ripperda,106,16 Freek Roelofs,4 Alan Rogers,12 Eduardo Ros,7 Mel Rose,20 Arash Roshanineshat,20 Helge Rottmann,7 Alan L. Roy,7 Chet Ruszczyk,12 Kazi L. J. Rygl,86 Salvador Sánchez,107 David Sánchez-Arguelles,66,67 Mahito Sasada,47,108 Tuomas Savolainen,109,110,7 F. Peter Schloerb,97 Karl-Friedrich Schuster,29 Lijing Shao,7,71 Zhiqiang Shen (沈志强),43,44 Des Small,79 Bong Won Sohn,33,34,111 Jason SooHoo,12 He Sun (孙赫),25 Fumie Tazaki,47 Alexandra J. Tetarenko,112 Paul Tiede,31,32 Remo P. J. Tilanus,4,64,113,20 Michael Titus,12 Kenji Toma,114,115 Pablo Torne,7,107 Tyler Trent,20 Efthalia Traianou,7 Sascha Trippe,116 Ilse van Bemmel,79 Huib Jan van Langevelde,79,117 Daniel R. van Rossum,4 Jan Wagner,7 Derek Ward-Thompson,118 John Wardle,119 Jonathan Weintroub,5,6 Norbert Wex,7 Robert Wharton,7 Maciek Wielgus,5,6 George N. Wong,62 Qingwen Wu (吴庆文),120 Doosoo Yoon,42 Andr´e Young,4 Ken Young,6 Feng Yuan (袁峰),43,68,121 Ye-Fei Yuan (袁业飞),122 J. Anton Zensus,7 Guang-Yao Zhao,14 and Shan-Shan Zhao43

(EHT Collaboration)

1Institut für Theoretische Physik, Goethe-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt, Germany 2Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt, Germany 3School of Mathematics, Trinity College, Dublin 2, Ireland 4Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box 9010, 6500 GL Nijmegen, Netherlands 5Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, Massachusetts 02138, USA

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6Center for Astrophysics—Harvard & Smithsonian, 60 Garden Street, Cambridge, Massachusetts 02138, USA 7Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany 8Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China 9Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany 10Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, GR 15783 Zografos, Greece 11Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, United Kingdom 12Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, Massachusetts 01886, USA 13National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 14Instituto de Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, E-18008 Granada, Spain 15Department of Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia 16Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA 17Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C. 18Departament d’Astronomia i Astrofísica, Universitat de Val`encia, C. Dr. Moliner 50, E-46100 Burjassot, Val`encia, Spain 19Observatori Astronòmic, Universitat de Val`encia, C. Catedrático Jos´e Beltrán 2, E-46980 Paterna, Val`encia, Spain 20Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Avenue, Tucson, Arizona 85721, USA 21Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, Illinois 60510, USA 22Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA 23East Asian Observatory, 660 N. A’ohoku Place, Hilo, Hawaii 96720, USA 24Nederlandse Onderzoekschool voor Astronomie (NOVA), PO Box 9513, 2300 RA Leiden, Netherlands 25California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA 26Institute of Astronomy and Astrophysics, Academia Sinica, 645 N. A’ohoku Place, Hilo, Hawaii 96720, USA 27Department of Physics, McGill University, 3600 rue University, Montr´eal, Quebec City H3A 2T8, Canada 28McGill Space Institute, McGill University, 3550 rue University, Montr´eal, Quebec City H3A 2A7, Canada 29Institut de Radioastronomie Millim´etrique, 300 rue de la Piscine, F-38406 Saint Martin d’H`eres, France 30Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada 31Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada 32Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 33Korea Astronomy and Space Science Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea 34University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea 35Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA 36Department of Physics, University of Chicago, 5720 South Ellis Avenue, Chicago, Illinois 60637, USA 37Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA 38Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, New Jersey 08544, USA 39NASA Hubble Fellowship Program, Einstein Fellow 40Data Science Institute, University of Arizona, 1230 N. Cherry Avenue, Tucson, Arizona 85721, USA 41Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA 42Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, Netherlands 43Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People’s Republic of China

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44Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, People’s Republic of China 45Physics Department, Fairfield University, 1073 North Benson Road, Fairfield, Connecticut 06824, USA 46Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden 47Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan 48Department of Astronomical Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 49Department of Astronomy and Columbia Astrophysics Laboratory, Columbia University, 550 W 120th Street, New York, New York 10027, USA 50Dipartimento di Fisica “E. Pancini”, Universitá di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy 51INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy 52Wits Centre for Astrophysics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg 2050, South Africa 53Department of Physics, University of Pretoria, Hatfield, Pretoria 0028, South Africa 54Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, Makhanda 6140, South Africa 55LESIA, Observatoire de Paris, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, 5 place Jules Janssen, 92195 Meudon, France 56National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, PR China 57University of Massachusetts Boston, 100 William T. Morrissey Boulevard, Boston, Massachusetts 02125, USA 58National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, Virginia 22903, USA 59Steward Observatory and Department of Astronomy, University of Arizona, 933 North Cherry Avenue, Tucson, Arizona 85721, USA 60Department of History of Science, Harvard University, Cambridge, Massachusetts 02138, USA 61Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 62Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801, USA 63Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, Illinois 61801, USA 64Leiden Observatory—Allegro, Leiden University, P.O. Box 9513, 2300 RA Leiden, Netherlands 65CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark 66Instituto Nacional de Astrofísica, Óptica y Electrónica. Apartado Postal 51 y 216, 72000. Puebla Pue., M´exico 67Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940, Ciudad de M´exico, M´exico 68Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, Shanghai 200030, People’s Republic of China 69NOVA Sub-mm Instrumentation Group, Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, Netherlands 70Department of Astronomy, School of Physics, Peking University, Beijing 100871, People’s Republic of China 71Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People’s Republic of China 72Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 73The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan 74Department of Statistical Science, The Graduate University for Advanced Studies (SOKENDAI), 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan 75Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan 76Institute for Astrophysical Research, Boston University, 725 Commonwealth New Jersey, Boston, Massachusetts 02215, USA 77Astronomical Institute, St. Petersburg University, Universitetskij pr., 28, Petrodvorets,198504 St.Petersburg, Russia 78Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 79Joint Institute for VLBI ERIC (JIVE), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, Netherlands

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80Kogakuin University of Technology & Engineering, Academic Support Center, 2665-1 Nakano, Hachioji, Tokyo 192-0015, Japan 81Physics Department, National Sun Yat-Sen University, No. 70, Lien-Hai Rd, Kaosiung City 80424, Taiwan, R.O.C 82National Optical Astronomy Observatory, 950 N. Cherry Avenue, Tucson, Arizona 85719, USA 83Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan District, Beijing, People’s Republic of China 84School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China 85Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, People’s Republic of China 86Italian ALMA Regional Centre, INAF-Istituto di Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy 87Department of Physics, National Taiwan University, No.1, Sect.4, Roosevelt Road., Taipei 10617, Taiwan, R.O.C 88Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de M´exico, Morelia 58089, M´exico 89Instituto de Astronomía, Universidad Nacional Autónoma de M´exico, CdMx 04510, M´exico 90Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan Province, People’s Republic of China 91Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, People’s Republic of China 92Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming, People’s Republic of China 93Gravitation Astroparticle Physics Amsterdam (GRAPPA) Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands 94School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA 95Astronomy Department, Universidad de Concepción, Casilla 160-C, Concepción, Chile 96National Institute of Technology, Hachinohe College, 16-1 Uwanotai, Tamonoki, Hachinohe City, Aomori 039-1192, Japan 97Department of Astronomy, University of Massachusetts, 01003, Amherst, Massachusetts, USA 98South African Radio Astronomy Observatory, Observatory 7925, Cape Town, South Africa 99Villanova University, Mendel Science Center Rm. 263B, 800 E Lancaster Avenue, Villanova Pennsylvania 19085 100Physics Department, Washington University CB 1105, St Louis, Missouri 63130, USA 101Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada 102Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario M5S 3H4, Canada 103Canadian Institute for Advanced Research, 180 Dundas St West, Toronto, Ontario M5G 1Z8, Canada 104Radio Astronomy Laboratory, University of California, Berkeley, California 94720, USA 105Department of Physics, National Taiwan Normal University, No. 88, Sec. 4, Tingzhou Road, Taipei 116, Taiwan, R.O.C. 106Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, New Jersey 08544, USA 107Instituto de Radioastronomía Milim´etrica, IRAM, Avenida Divina Pastora 7, Local 20, E-18012, Granada, Spain 108Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 109Aalto University Department of Electronics and Nanoengineering, PL 15500, FI-00076 Aalto, Finland 110Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland 111Department of Astronomy, Yonsei University, Yonsei-ro 50, Seodaemun-gu, 03722 Seoul, Republic of Korea 112East Asian Observatory, 660 North A’ohoku Place, Hilo, Hawaii 96720, USA 113Netherlands Organisation for Scientific Research (NWO), Postbus 93138, 2509 AC Den Haag, Netherlands 114Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan 115Astronomical Institute, Tohoku University, Sendai 980-8578, Japan

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116Department of Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea 117Leiden Observatory, Leiden University, Postbus 2300, 9513 RA Leiden, Netherlands 118Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, United Kingdom 119Physics Department, Brandeis University, 415 South Street, Waltham, Massachusetts 02453, USA 120School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China 121School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, People’s Republic of China 122Astronomy Department, University of Science and Technology of China, Hefei 230026, People’s Republic of China

(Received 29 November 2020; accepted 21 April 2021; published 20 May 2021)

Our understanding of strong gravity near supermassive compact objects has recently improved thanks to the measurements made by the Event Horizon Telescope (EHT). We use here the M87* shadow size to infer constraints on the physical charges of a large variety of nonrotating or rotating black holes. For example, we show that the quality of the measurements is already sufficient to rule out that M87* is a highly charged dilaton black hole. Similarly, when considering black holes with two physical and independent charges, we are able to exclude considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes.

DOI: 10.1103/PhysRevD.103.104047

I. INTRODUCTION violations of the equivalence principle occur in strong- fields due, e.g., to the presence of additional dynamical General relativity (GR) was formulated to consistently fields, such as scalar [7,8] or vector fields [9–13], that may account for the interaction of dynamical gravitational fields fall off asymptotically. Agreement with the predictions of with matter and energy, the central idea of which is that the GR coming from observations of binary pulsars [14–16], former manifests itself through modifications of spacetime and of the gravitational redshift [17] and geodetic orbit- geometry and is fully characterized by a metric tensor. precession [18] of the star S2 near our galaxy’s central While the physical axioms that GR is founded on are supermassive compact object Sgr A⋆ by the GRAVITY contained in the equivalence principle [1,2], the Einstein- collaboration, all indicate the success of GR in describing Hilbert action further postulates that the associated strong-field physics as well. In addition, with the gravita- equations of motion involve no more than second-order tional-wave detections of coalescing binaries of compact derivatives of the metric tensor. objects by the LIGO/Virgo collaboration [19,20] and the The strength of the gravitational field outside an object of first images of black holes produced by EHT, it is now mass M and characteristic size R, in geometrized units possible to envision testing GR at the strongest field (G ¼ c ¼ 1), is related to its compactness C ≔ M=R, strengths possible. which is ∼10−6 for the Sun, and takes values ∼0.2–1 for While the inferred size of the shadow from the recently compact objects such as neutron stars and black holes. obtained horizon-scale images of the supermassive com- Predictions from GR have been tested and validated by pact object in M87 galaxy by the EHT collaboration [21– various solar-system experiments to very high precision 26] was found to be consistent to within 17% for a [2,3], setting it on firm footing as the best-tested theory of 68% confidence interval of the size predicted from GR classical gravity in the weak-field regime. It is important, for a Schwarzschild black hole using the a priori known however, to consider whether signatures of deviations from estimates for the mass and distance of M87* based on the Einstein-Hilbert action, e.g., due to higher derivative stellar dynamics [27], this measurement admits other terms [4–6], could appear in measurements of phenomena possibilities, as do various weak-field tests [2,28]. Since occurring in strong-field regimes where C is large. the number of alternative theories to be tested using this Similarly, tests are needed to assess whether generic measurement is large, a systematic study of the constraints set by a strong-field measurement is naturally more tractable within a theory-agnostic framework, and various Published by the American Physical Society under the terms of such systems have recently been explored [29–36]. This the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to approach allows for tests of a broad range of possibilities the author(s) and the published article’s title, journal citation, that may not be captured in the limited set of known and DOI. solutions. This was exploited in Ref. [28], where

104047-5 PRASHANT KOCHERLAKOTA et al. PHYS. REV. D 103, 104047 (2021) constraints on two deformed metrics were obtained when black-hole solutions [72–75] which can be expressed in the determining how different M87* could be from a Kerr Newman-Janis form [76] to facilitate straightforward ana- black hole while remaining consistent with the EHT lytical computations. It is to be noted that this study is meant measurements. to be a proof of principle and that while the constraints we can However, because such parametric tests cannot be set here are limited, the analytical procedure outlined below connected directly to an underlying property of the alter- for this large class of metrics is general, so that as future native theory, here we use instead the EHT measurements observations become available, we expect the constraints that to set bounds on the physical parameters, i.e., angular can be imposed following the approach proposed here to be momentum, electric charge, scalar charge, etc.—and which much stronger. we will generically refer to as “charges” (or hairs)—that various well-known black-hole solutions depend upon. II. SPHERICAL NULL GEODESICS AND – Such analyses can be very instructive [37 51] since they SHADOWS can shed light on which underlying theories are promising candidates and which must be discarded or modified. At the For all the static, spherically symmetric spacetimes we same time, they may provide insight into the types of consider here, the definition of the shadow can be cast in additional dynamical fields that may be necessary for a rather general terms. In particular, for all the solutions complete theoretical description of physical phenomena, considered, the line element expressed in areal-radial polar 1 and whether associated violations of the equivalence coordinates ðt; r;˜ θ; ϕÞ has the form principle occur. More specifically, since the bending of light in the 2 μ ν 2 gðr˜Þ 2 2 2 ds ¼ gμνdx dx ¼ −fðr˜Þdt þ dr˜ þ r˜ dΩ ; ð1Þ presence of curvature—either in static or in stationary fðr˜Þ 2 spacetimes—is assured in any metric theory of gravity, and the presence of large amounts of mass in very small and the photon region, which degenerates into a photon volumes can allow for the existence of a region where sphere, is located at r˜ ≕ r˜ , which can be obtained by null geodesics move on spherical orbits, an examination of ps solving [28] the characteristics of such photon regions, when they exist, is a useful first step. The projected asymptotic collection of 2fðr˜Þ the photons trajectories that are captured by the black hole r˜ − ¼ 0: ð2Þ —namely, all of the photon trajectories falling within the ∂r˜fðr˜Þ value of the impact parameter at the unstable circular orbit in the case of nonrotating black holes—will appear as a The boundary of this photon sphere when observed from dark area to a distant observer and thus represents the the frame of an asymptotic observer, due to gravitational “shadow” of the capturing compact object. This shadow— lensing, appears to be a circle of size [28] which can obviously be associated with black holes [52– 57], but also more exotic compact objects such as grav- r˜ ˜ ¼ pffiffiffiffiffiffiffiffiffiffiffiffips ð Þ astars [58,59] or naked singularities [60,61]—is determined rsh : 3 fðr˜ Þ entirely by the underlying spacetime metric. Therefore, the ps properties of the shadow—and at lowest order its size— represent valuable observables common to all metric On the other hand, the class of Newman-Janis stationary, theories of gravity, and can be used to test them for their axisymmetric spacetimes we consider here [76], which are agreement with EHT measurements. geodesically integrable (see, e.g., [55,77,78]), can be While the EHT measurement contains far more informa- expressed in Boyer-Lindquist coordinates (t; r; θ; ϕ)as tion related to the flow of magnetized plasma near M87*, we will consider only the measurement of the size of the bright ds2 ¼ −fdt2 − 2asin2θð1 − fÞdtdϕ ring. Here we consider various spherically symmetric black- Σ þ½Σ þ 2 2θð2 − Þ 2θ ϕ2 þ 2 þ Σ θ2 hole solutions, from GR that are either singular (see, e.g., a sin f sin d Δ dr d ; [62]) or non-singular [63–65], and string theory [66–70]. Additionally, we also consider the Reissner-Nordström (RN) ð4Þ and the Janis-Newman-Winicour (JNW) [71] naked singu- larity solutions, the latter being a solution of the Einstein- where f ¼ fðr; θÞ and Σðr; θÞ ≔ r2 þ a2 cos2 θ and Klein-Gordon system. Many of these solutions have been ΔðrÞ ≔ Σðr; θÞfðr; θÞþa2 sin2 θ. In particular, these are recently summarized in Ref. [36], where they were cast in a generalized expansion of static and spherically symmetric 1We use the tilde on the radial coordinate of static spacetimes metrics. Since angular momentum plays a key role in to distinguish it from the corresponding radial coordinate of astrophysical scenarios, we also consider various rotating axisymmetric spacetimes.

104047-6 CONSTRAINTS ON BLACK-HOLE CHARGES WITH THE 2017 … PHYS. REV. D 103, 104047 (2021) the stationary generalizations obtained by employing the since its image, as seen by an asymptotic observer, is the Newman-Janis algorithm [76]) for “seed” metrics of the shadow. We denote the inner and outer surfaces of this ð˜Þ¼1 2 ðθÞ ðθÞ form (1) with g r . photon region by rp− and rpþ respectively, with the The Lagrangian L for geodesic motion in the spacetime former (smaller) SNG corresponding to the location of a 2L ≔ _μ _ν ξ ð Þ 0 (4) is given as gμνx x , where an overdot represents a prograde photon orbit (i.e., SNG rp− > ), and the latter to derivative with respect to the affine parameter, and 2L ¼ a retrograde orbit. −1 for timelike geodesics and 2L ¼ 0 for null geodesics. It can be shown that all of the SNGs that are ∂ ∂ The two Killing vectors t and ϕ yield two constants of admitted in the photon region, for both the spherically motion symmetric and axisymmetric solutions considered here, are unstable to radial perturbations. In particular, for 2 _ −E ¼ −ft_ − a sin θð1 − fÞϕ; the stationary solutions, the stability of SNGs at a ¼ L ¼ −a sin2 θð1 − fÞt_ þ½Σ þ a2 sin2 θð2 − fÞ sin2 θϕ_ ; radius r rSNG with respect to radial perturbations is determined by the sign of ∂2R, and when ∂2Rðr Þ > ð Þ r r SNG 5 0, SNGs at that radius are unstable. The expression for ∂2R reads in terms of which the geodesic equation for photons can be r separated into 8r ∂2R ¼ ½rð∂ ΔÞ2 − 2rΔ∂2Δ þ 2Δ∂ Δ: ð9Þ 2 2 2 2 2 r ð∂ ΔÞ2 r r r Σ r_ ¼ðr þ a − aξÞ − ΔI ≕ RðrÞ; ð6Þ r

2 Σ2θ_ ¼ I − ða sin θ − ξ csc θÞ2 ≕ ΘðθÞ; ð7Þ To determine the appearance of the photon region and the associated shadow, as seen by asymptotic observers, where we have introduced first ξ ≔ L=E, and then we can introduce the usual notion of celestial coordinates I ≔ η þða − ξÞ2. Also, η is the Carter constant, and the ðα; βÞ, which for any photon with constants of the motion existence of this fourth constant of motion is typically ðξ; ηÞ can be obtained, for an asymptotic observer present associated with the existence of an additional Killing-Yano at an inclination angle i with respect to the spin-axis of tensor (see for example [56,80]). the compact object as in [82]. For photons on an SNG, In particular, we are interested here in spherical null we can set the conserved quantities (ξ, η) to the values geodesics (SNGs), which satisfy r_ ¼ 0 and ̈r ¼ 0 and are given in Eq. (8) above to obtain [80,81] not necessarily planar; equivalently, SNGs can exist at Rð Þ¼0 Rð Þ ¼ 0 locations where r and d r =dr . Since these α ¼ −ξ csc i; ð10Þ are only two equations in three variables (r, ξ, η), it is sh SNG convenient, for reasons that will become evident below, to obtain the associated conserved quantities along such β ¼ðη þ 2 2 − ξ2 2 Þ1=2 ð Þ sh SNG a cos i SNGcot i : 11 SNGs in terms of their radii r as (see also [81]), pffiffiffiffiffiffiffiffiffi 2 þ 2 4 Δ Recognizing that β ¼ ΘðiÞ, it becomes clear that only ξ ð Þ¼r a − r SNG r ; the SNGs with ΘðiÞ ≥ 0 determine the apparent shadow a a∂rΔ shape. Since the photon region is not spherically symmetric r2 η ðrÞ¼ ½16a2Δ − ðr∂ Δ − 4ΔÞ2: ð Þ in rotating spacetimes, the associated shadow is also not SNG 2ð∂ ΔÞ2 r 8 a r circular in general. It can be shown that the band of radii for which SNGs can exist narrows as we move away from the The condition that ΘðθÞ ≥ 0, which must necessarily hold equatorial plane, and reduces to a singlevalue at the pole, i.e., as can be seen from Eq. (7), restricts the radial range for in the limit θ → π=2,wehaver þ ¼ r − (see e.g., Fig. 3.3 which SNGs exist, and it is evident that this range depends p p on θ. This region, which is filled by such SNGs, is called of [52]). As a result, the parametric curve of the shadow the photon region (see, e.g., Fig. 3.3 of [52]). boundary as seen by an asymptotic observer lying along the α2 þ β2 ¼ η ð Þþ The equality ΘðθÞ¼0 determines the boundaries of the pole is perfectly circular, sh sh SNG rp;π=2 ξ2 ð Þ photon region, and the (disconnected) piece which lies in SNG rp;π=2 . the exterior of the outermost horizon is of primary interest We can now define the characteristic areal-radius of the shadow curve as [83] 2 Note that while the Sen solution can be obtained via the  Z  1 2 Newman-Janis algorithm [79], the starting point is the static 2 rpþ = ρ ≔ β ð Þ∂ α ð Þ ð Þ EMd-1 metric written in a non-areal-radial coordinate such that rsh;A π dr sh r r sh r : 12 r − gttgρρ ¼ −1. p

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III. SHADOW SIZE CONSTRAINTS FROM THE radius estimate from stellar dynamics yields a prediction for θ ¼ θ ¼ 2017 EHT OBSERVATIONS OF M87* thepffiffiffi angular shadow radius sh rsh=D as being sh 3 3θ Measurements of stellar dynamics near M87* were dyn. The 2017 EHT measurement, which includes previously used to produce a posterior distribution function spin-dependent effects as described above and which of the angular gravitational radius θ ≔ M=D, where M is probes near-horizon scales, then determines the allowed g θ ≈ the mass of and D the distance to M87*. The 2017 EHT spreadpffiffiffi in the angular shadow diameter as, sh 3 3ð1 0 17Þθ observations of M87* can be similarly used to determine . g, at 68% confidence levels [28]. Finally, θ θ such a posterior [26]. These observations were used to since both angular estimates sh and g make use of the determine the angular diameter dˆ of the bright emission same distance estimate to M87*, it is possible to convert the 1 σ θ ring that surrounds the shadow [26]. In Sec. 5.3 there, using - bounds on sh to bounds on the allowed shadow size synthetic images from general-relativistic magnetohydro- for M87*. dynamics (GRMHD) simulations of accreting Kerr black That is, independently of whether the underlying sol- holes for a wide range of physical scenarios, the scaling ution be spherically symmetric (in which case we will ˜ ð Þ factor α ¼ d=ˆ θ was calibrated. For emission from the consider rsh) or axisymmetric prshffiffiffi;A , the shadow size of g 3 3ð1 0 17Þ outermost boundary of the photon region of a Kerr black M87* must lie in the range . M [28], i.e., hole, α should lie in the range ≃9.6–10.4. (see gray-shaded region in Fig. 2) The EHT measurement picks out a class of best-fit 4 31 ≈ ≤ ˜ ≤ ≈ 6 08 images (“top-set”) from the image library, with a mean . M rsh;EHT- min rsh;rsh;A rsh;EHT- max . M; value for α of 11.55 (for the “xs-ring” model) and 11.50 ð14Þ (for the “xs-ringauss” model), when using two different geometric crescent models for the images, implying that the where we have introduced the maximum/minimum geometric models were accounting for emission in the top- shadow radii rsh;EHT- max=rsh;EHT- min inferred by the EHT, set GRMHD images that preferentially fell outside of the at 68% confidence levels. photon ring. Using the distribution of α for these top-set Note that the bounds thus derived are consistent with images then enabled the determination of the posterior in compact objects that cast shadows that are both signifi- ðθ Þ the angular gravitational radius Pobs g for the EHT data. cantly smaller and larger than the minimum and maximum It is to be noted that this posterior was also determined shadow sizes that a Kerr black hole could cast, which lie in using direct GRMHD fitting, and image domain feature the range, 4.83M − 5.20M (see, e.g., [28,84]). extraction procedures, as described in Sec. 9.2 there, and a An important caveat here is that the EHT posterior θ high level of consistency was found across all measurement distribution on g was obtained after a comparison with a methods. Finally, in Sec. 9.5 of [26], the fractional large library of synthetic images built from GRMHD deviation in the angular gravitational radius δ was intro- simulations of accreting Kerr black holes [25]. Ideally, a duced as rigorous comparison with non-Kerr solutions would require a similar analysis and posterior distributions built from θ equivalent libraries obtained from GRMHD simulations of δ ≔ g − 1 ð Þ θ ; 13 such non-Kerr solutions. Besides being computationally dyn unfeasible, this approach is arguably not necessary in θ θ practice. For example, the recent comparative analysis of where g and dyn were used to denote the EHT and the stellar-dynamics inferences of the angular gravitational Ref. [50] has shown that the image libraries produced in radius, respectively. The posterior on δ—as defined in this way would be very similar and essentially indistin- — guishable, given the present quality of the observations. As Eq. (32) of [26] was then obtained (see Fig. 21 there), and 1 σ its width was found to be δ ¼ −0.01 0.17, for a 68% a result, we adopt here the working assumption that the - credible interval. This agreement of the 2017 EHT meas- uncertainty in the shadow angular size for non-Kerr solutions is very similar to that for Kerr black holes, and urement of the angular gravitational radius for M87* with a hence employ the constraints (14) for all of the solutions previously existing estimate for the same, at much larger considered here. distances, constitutes a validation of the null hypothesis of the EHT, and in particular that M87* can be described by IV. NOTABLE PROPERTIES OF VARIOUS the Kerr black-hole solution. SPACETIMES Since the stellar dynamics measurements [27] are sensi- tive only to the monopole of the metric (i.e., the mass) due As mentioned above, a rigorous comparison with non- to negligible spin-dependent effects at the distances Kerr black holes would require constructing a series of involved in that analysis, modeling M87* conservatively exhaustive libraries of synthetic images obtained from using the Schwarzschild solution is reasonable with their GRMHD simulations on such non-Kerr black holes. In obtained posterior. Then, using the angular gravitational turn, this would provide consistent posterior distributions

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TABLE I. Summary of properties of spacetimes used here. For easy access, we show whether the spacetime contains a rotating compact object or not, whether it contains a spacetime singularity, and what type of stationary nongravitational fields are present in the spacetime. Starred spacetimes contain naked singularities and daggers indicate a violation of the equivalence principle (see, e.g., [36]); In particular, these indicate violations of the weak equivalence principle due to a varying fine structure constant, a result of the coupling of the dilaton to the EM Lagrangian [36,89].

Spacetime Rotation Singularity Spacetime content KN [73] Yes Yes EM fields Kerr [72] Yes Yes vacuum RN [62] No Yes EM fields RN* [62] No Yes EM fields Schwarzschild [62] No Yes vacuum Rot. Bardeen [75] Yes No matter Bardeen [63] No No matter Rot. Hayward [75] Yes No matter Frolov [65] No No EM fields, matter Hayward [64] No No matter JNW* [71] No Yes scalar field KS [66] No Yes vacuum Sen† [74] Yes Yes EM, dilaton, axion fields EMd-1† [67,68] No Yes EM, dilaton fields EMd-2† [70] No Yes EM, EM, dilaton fields of angular gravitational radii for the various black holes and the non-vacuum spacetimes used here is of the order of the hence determine how δ varies across different non-Kerr mass of the central compact object M, while the total mass black holes, e.g., for Sen black holes. Because this is of the accreting plasma in the GRMHD simulations is only computationally unfeasible—the construction of only the a tiny fraction of the same, it is reasonable to treat the Kerr library has required the joint effort of several groups spacetime geometry and the stationary fields as unaffected with the EHTC over a good fraction of a year—we briefly by the plasma. Fourth, we have also been careful not to use discuss below qualitative arguments to support our use of solutions from theories with modified electrodynamics the bounds given in Eq. (14) above as an approximate, yet (such as nonlinear electrodynamics). As a result, the indicative, measure. electromagnetic Lagrangian in all of the theories consid- To this end, we summarize in Table I the relevant ered here is the Maxwell Lagrangian (see, e.g., the properties of the various solutions used here. First, we discussion in [36] and compare with [53]). This ensures have considered here solutions from three types that in these spacetimes light moves along the null geo- of theories, i.e., the underlying actions are either desics of the metric tensor (see, e.g., Sec. 4.3 of [62] and (a) Einstein-Hilbert-Maxwell-matter [62–66,71,72,75], compare against Sec. 2 of [88]). Therefore, we are also (b) Einstein-Hilbert-Maxwell-dilaton-axion [67,68,74],or assured that ray-tracing the radiation emitted from the (c) Einstein-Hilbert-Maxwell-Maxwell-dilaton [70]. This accreting matter in these spacetimes can be handled careful choice implies that the gravitational piece of the similarly as in the Kerr spacetime. action is always given by the Einstein-Hilbert term and that Finally, under the assumption that the dominant effects matter is minimally coupled to gravity. As a result, the in determining the angular gravitational radii come from dynamical evolution of the accreting plasma is expected to variations in the location of the photon region and in be very similar to that in GR, as indeed found in Ref. [50]. location of the inner edge of the accretion disk in these Second, since a microphysical description that allows one spacetimes, it is instructive to learn how these two physical to describe the interaction of the exotic matter present in quantities vary when changing physical charges, and, in some of the regular black-hole spacetimes used here particular, to demonstrate that they are quantitatively [63,64]—which typically do not satisfy some form of comparable to the corresponding values for the Kerr the energy conditions [75,85]—with the ordinary matter spacetime. is thus far lacking, it is reasonable to assume that the For this purpose, we study the single-charge solutions interaction between these two types of fluids is gravita- used here and report in Fig. 1 the variation in the location of tional only. This is indeed what is done in standard the photon spheres (left panel) and innermost stable circular numerical simulations, either in dynamical spacetimes orbit (ISCO) radii (right panel) as a function of the relevant (see, e.g., [86]), or in fixed ones [49,87]. Third, since physical charge (cf. left panel of Fig. 1 in the main text). the mass-energy in the matter and electromagnetic fields for Note that both the photon-sphere and the ISCO radii

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FIG. 1. Left: variation in the photon sphere radii for the single-charge nonrotating solutions as a function of the normalized physical charge. Right: The same as in the left panel but for the ISCO radii. We include also, for comparison, the variation in the Kerr equatorial prograde and retrograde photon sphere and ISCO radii in the left and right panels respectively.

depend exclusively on the gtt component of the metric solutions given by Reissner-Nordström (RN) [62], when expressed using an areal radial coordinate r˜ (see, e.g., Bardeen [63,75], Hayward [64,91], Kazakov-Solodhukin [28,36]). To gauge the effect of spin, we also show the (KS) [66], and also the asymptotically-flat Einstein- variation in the locations of the equatorial prograde and Maxwell-dilaton (EMd-1) with ϕ∞ ¼ 0 and α1 ¼ 1 retrograde circular photon orbits and the ISCOs in the Kerr [67,68,88] solution (see Sec. IV of [36] for further details black-hole spacetime, expressed in terms of the Cartesian on these solutions). For each of these solutions we vary Kerr-Schild radial coordinate rCKS, which, in the equatorial the corresponding charge (in units of M) in thep allowedffiffiffiffiffiffiffiffiffiffiffiffiffi plane, is related to the Boyer-Lindquist radial coordinate 0 ¯ ≤ 1 0 ¯ ≤ 16 27 range, i.e., RN: naked singularity is entirely eliminated as a viable model for M87* and the Janis-Newman-Winicour naked singularity 3 parameter space is restricted further by this measurement to While the electromagnetic and scalar charge parameters for ˆ the RN and JNW spacetimes are allowed to take values q>¯ 1 and 0 < ν¯ ≲ 0.47. Finally, we also find that the KS black hole is ˆ ¯ 1 53 p0 <ffiffiffiffiffiffiffiffiν¯ ≔ 1 − ν¯ < 1 respectively, they do not cast shadows for q>¯ also restricted to have charges in the range l< . .In 9=8 and 0.5 ≤ ν¯ˆ < 1 (see, e.g., Sec. IV D of [36] and addition, note that the nonrotating Einstein-Maxwell- references therein). dilaton 2 (EMd-2) solution [70]—which depends on two

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˜ FIG. 2. Left: shadow radii rsh for various spherically symmetric black-hole solutions, as well as for the JNW and RN naked singularities (marked with an asterisk), as a function of the physical charge normalized to its maximum value. The gray/red shaded regions refer to the areas that are 1-σ consistent/inconsistent with the 2017 EHT observations and highlight that the latter set constraints on the physical charges (see also Fig. 3 for the EMd-2 black hole). Right: shadow areal radii rsh;A as a function of the dimensionless spin a for four families of black-hole solutions when viewed on the equatorial plane (i ¼ π=2). Also in this case, the observations restrict the ranges of the physical charges of the Kerr-Newman and the Sen black holes (see also Fig. 3). independent charges—can also produce shadow radii that are To further explore the constraints on the excluded incompatible with the EHTobservations; we will discuss this regions for the Einstein-Maxwell-dilaton 2 and the Sen further below. The two EMd black-hole solutions (1 and 2) black holes, we report in Fig. 3 the relevant ranges for these correspond to fundamentally different field contents, as two solutions. The Einstein-Maxwell-dilaton 2 black holes discussed in [70]. are nonrotating but have twopffiffiffi physical chargesp expressedffiffiffi by ¯ ¯ We report in the right panel of Fig. 2 the shadow the coefficients 0 < qe < 2 and 0 < qm < 2, while the areal radius rsh;A for a number of stationary black holes, Sen black holes spin (a) and have an additional electro- ¯ such as Kerr [72], Kerr-Newman (KN) [73],Sen[74], magnetic charge qm. Also in this case, the gray/red shaded and the rotating versions of the Bardeen and Hayward regions refer to the areas that are consistent/inconsistent black holes [75]. The data refers to an observer with the 2017 EHT observations. The figure shows rather inclination angle of i ¼ π=2, and we find that the easily that for these two black-hole families there are large variation in the shadow size with spin at higher inclinations (of up to i ¼ π=100) is at most about 7.1% (for i ¼ π=2, this is 5%); of course, at zero-spin the shadow size does not change with inclination. The shadow areal radii are shown as a function of the dimensionless spin of the black hole a ≔ J=M2,where J is its angular momentum, and for representative values of the additional parameters that characterize the solu- tions. Note that—similar to the angular momentum for a Kerr black hole—theroleofanelectricchargeorthe presenceofadeSittercore(asinthecaseofthe Hayward black holes) is to reduce the apparent size of the shadow. Furthermore, on increasing the spin para- meter, we recover the typical trend that the shadow becomes increasingly noncircular, as encoded, e.g., in δ the distortion parameter sh defined in [57,83] (see Appendix). Also in this case, while the regular rotating Bardeen and Hayward solutions are compatible with the FIG. 3. Constraints set by the 2017 EHT observations on the present constraints set by the 2017 EHT observations, nonrotating Einstein-Maxwell-dilaton 2 and on the rotating Sen the Kerr-Newman and Sen families of black holes can black holes. Also in this case, the gray/red shaded regions refer to produce shadow areal radii that lie outside of the 1-σ the areas that are 1-σ consistent/inconsistent with the 2017 EHT region allowed by the observations. observations).

104047-11 PRASHANT KOCHERLAKOTA et al. PHYS. REV. D 103, 104047 (2021) areas of the space of parameters that are excluded at the 1-σ size induced by mass and spin remains and is inevitable, level. Not surprisingly, these areas are those where the when in the future the relative difference in the posterior for physical charges take their largest values and hence the the angular gravitational radius for M87* can be pushed to corresponding black-hole solutions are furthest away from ≲5%, we should be able to constrain its spin, when the corresponding Schwarzschild or Kerr solutions. The modeling it as a Kerr black hole. Furthermore, since this obvious prospect is of course that as the EHT increases the posterior implies a spread in the estimated mass, one can precision of its measurements, increasingly larger portions expect small changes in the exact values of the maximum of the space of parameters of these black holes will be allowed charges reported here. Hence, as future observa- excluded. Furthermore, other solutions that are presently tions—either in terms of black-hole imaging or of gravi- still compatible with the observations may see their tational-wave detection—will become more precise and corresponding physical charges restricted. notwithstanding a poor measurement of the black-hole spin, the methodology presented here can be readily VI. CONCLUSIONS applied to set even tighter constraints on the physical charges of non-Einsteinian black holes. As our understanding of gravity under extreme regimes improves, and as physical measurements of these regimes ACKNOWLEDGMENTS are now becoming available—either through the imaging of supermassive black holes or the detection of gravitational It is a pleasure to thank Enrico Barausse, Sebastian waves from stellar-mass black holes—we are finally in the Völkel and Nicola Franchini for insightful discussions on position of setting some constraints to the large landscape alternative black holes. Support comes the ERC Synergy of non-Kerr black holes that have been proposed over the Grant “BlackHoleCam: Imaging the Event Horizon of years. We have used here the recent 2017 EHT observations Black Holes” (Grant No. 610058). During the completion of M87* to set constraints, at the 1-σ-level, on the physical of this work we have become aware of a related work by charges—either electric, scalar, or angular momentum—of S. Völkel et al. [98], which deals with topics that partly a large variety of static (nonrotating) or stationary (rotating) overlap with those of this manuscript (i.e., EHT tests of the black holes. strong-field regime of GR). The authors of the present In this way, when considering nonrotating black holes paper thank the following organizations and programs: the with a single physical charge, we have been able to rule out, Academy of Finland (projects No. 274477, No. 284495, at 68% confidence levels, the possibility that M87* is a No. 312496, No. 315721); the Alexander von Humboldt near-extremal Reissner-Nordström or Einstein-Maxwell- Stiftung; Agencia Nacional de Investigación y Desarrollo dilaton 1 black hole and that the corresponding physical (ANID), Chile via NCN19\_058 (TITANs), and Fondecyt charge must be in the range, RN: 0 < q¯ ≲ 0.90 and EMd-1: 3190878; an Alfred P. Sloan Research Fellowship; Allegro, 0 < q¯ ≲ 0.95. We also find that it cannot be a Reissner- the European ALMA Regional Centre node in the Nordström naked singularity or a JNW naked singularity Netherlands, the NL astronomy research network NOVA with large scalar charge, i.e., only 0 < ν¯ˆ ≲ 0.47 is allowed. and the astronomy institutes of the University of Similarly, when considering black holes with two physical Amsterdam, Leiden University and Radboud University; charges (either nonrotating or rotating), we have been able the Black Hole Initiative at Harvard University, through a to exclude, with 68% confidence, considerable regions of grant (60477) from the John Templeton Foundation; the the space of parameters in the case of the Einstein- China Scholarship Council; Consejo Nacional de Ciencia y Maxwell-dilaton 2, Kerr-Newman and Sen black holes. Tecnología (CONACYT, Mexico, projects No. U0004- Although the idea of setting such constraints is an old one 246083, No. U0004-259839, No. F0003-272050, (see, e.g., [29–36,51,54,55]), and while there have been No. M0037-279006, No. F0003-281692, No. 104497, recent important developments in the study of other No. 275201, No. 263356); the Delaney Family via the possible observational signatures of such alternative sol- Delaney Family John A.Wheeler Chair at Perimeter utions, such as in X-ray spectra of accreting black holes Institute; Dirección General de Asuntos del Personal (see, e.g., [92]) and in gravitational waves [88,93–97],to Acad´emico-Universidad Nacional Autónomade M´exico the best of our knowledge, constraints of this type have not (DGAPA-UNAM, projects No. IN112417 and been set before for the spacetimes considered here. No. IN112820); the EACOA Fellowship of the East As a final remark, we note that while we have chosen Asia Core Observatories Association; the European only a few solutions that can be seen as deviations from the Research Council Synergy Grant “BlackHoleCam: Schwarzschild/Kerr solutions since they share the same Imaging the Event Horizon of Black Holes” (Grant basic Einstein-Hilbert-Maxwell action of GR, the work No. 610058); the Generalitat Valenciana postdoctoral presented here is meant largely as a proof-of-concept grant APOSTD/2018/177 and GenT Program (project investigation and a methodological example of how to No. CIDEGENT/2018/021); MICINN Research Project exploit observations and measurements that impact the No. PID2019-108995GB-C22; the Gordon and photon region. While a certain degeneracy in the shadow Betty Moore Foundation (Grants No. GBMF-3561,

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No. GBMF-5278); the Istituto Nazionale di Fisica Nucleare No. 11721303, No. 11725312, No. 11933007, (INFN) sezione di Napoli, iniziative specifiche No. 11991052, No. 11991053); a fellowship of China TEONGRAV; the International Max Planck Research Postdoctoral Science Foundation (2020M671266); the School for Astronomy and Astrophysics at the Natural Sciences and Engineering Research Council of Universities of Bonn and Cologne; Joint Princeton/ Canada (NSERC, including a Discovery Grant and the Flatiron and Joint Columbia/Flatiron Postdoctoral NSERC Alexander Graham Bell Canada Graduate Fellowships, research at the Flatiron Institute is supported Scholarships-Doctoral Program); the National Research by the Simons Foundation; the Japanese Government Foundation of Korea (the Global PhD Fellowship Grant: (Monbukagakusho: MEXT) Scholarship; the Japan grants No. 2014H1A2A1018695, No. NRF- Society for the Promotion of Science (JSPS) Grant-in- 2015H1A2A1033752, No. 2015-R1D1A1A01056807, Aid for JSPS Research Fellowship (JP17J08829); the Key the Korea Research Fellowship Program: NRF- Research Program of Frontier Sciences, Chinese Academy 2015H1D3A1066561, Basic Research Support Grant of Sciences (CAS, grants No. QYZDJ-SSW-SLH057, No. 2019R1F1A1059721); the Netherlands Organization No. QYZDJSSW-SYS008, No. ZDBS-LY-SLH011); the for Scientific Research (NWO) VICI award (Grant Lever-hulme Trust Early Career Research Fellowship; the No. 639.043.513) and Spinoza Prize SPI 78-409; the Max-Planck-Gesellschaft (MPG); the Max Planck Partner New Scientific Frontiers with Precision Radio Group of the MPG and the CAS; the MEXT/JSPS Interferometry Fellowship awarded by the South African KAKENHI (Grants No. 18KK0090, No. JP18K13594, Radio Astronomy Observatory (SARAO), which is a No. JP18K03656, No. JP18H03721, No. 18K03709, facility of the National Research Foundation (NRF), an No. 18H01245, No. 25120007); the Malaysian agency of the Department of Science and Innovation (DSI) Fundamental Research Grant Scheme (FRGS) FRGS/1/ of South Africa; the South African Research Chairs 2019/STG02/UM/02/6; the MIT International Science and Initiative of the Department of Science and Innovation Technology Initiatives (MISTI) Funds; the Ministry of and National Research Foundation; the Onsala Space Science and Technology (MOST) of Taiwan (105-2112- Observatory (OSO) national infrastructure, for the provi- M-001-025-MY3, 106-2112-M-001-011, 106-2119-M- sioning of its facilities/observational support (OSO receives 001-027, 107-2119-M-001-017, 107-2119-M-001-020, funding through the Swedish Research Council under 107-2119-M-110-005,108-2112-M-001-048, and 109- Grant No. 2017-00648) the Perimeter Institute for 2124-M-001-005); the National Aeronautics and Space Theoretical Physics (research at Perimeter Institute is Administration (NASA, Fermi Guest Investigator supported by the Government of Canada through the grant 80NSSC20K1567 and 80NSSC20K1567, NASA Department of Innovation, Science and Economic Astrophysics Theory Program Grant Development and by the Province of Ontario through No. 80NSSC20K0527, NASA NuSTAR Award the Ministry of Research, Innovation and Science); the No. 80NSSC20K0645, NASA Grant Spanish Ministerio de Ciencia e Innovación (grants No. NNX17AL82G, and Hubble Fellowship Grant No. PGC2018-098915-B-C21, No. AYA2016-80889-P; No. HST-HF2-51431.001-A awarded by the Space No. PID2019-108995GB-C21, No. PGC2018-098915-B- Telescope Science Institute, which is operated by the C21); the State Agency for Research of the Spanish MCIU Association of Universities for Research in Astronomy, through the “Center of Excellence Severo Ochoa” award Inc., for NASA, under contract NAS5-26555); the National for the Instituto de Astrofísica de Andalucía (SEV-2017- Institute of Natural Sciences (NINS) of Japan; the National 0709); the Toray Science Foundation; the Consejería de Key Research and Development Program of China (Grants Economía, Conocimiento, Empresas y Universidad, Junta No. 2016YFA0400704, No. 2016YFA0400702); the de Andalucía (Grant No. P18-FR-1769), the Consejo National Science Foundation (NSF, Grants No. AST- Superior de Investigaciones Científicas (Grant 0096454, No. AST-0352953, No. AST-0521233, No. 2019AEP112); the U.S. Department of Energy No. AST-0705062, No. AST-0905844, No. AST- (USDOE) through the Los Alamos National Laboratory 0922984, No. AST-1126433, No. AST-1140030, (operated by Triad National Security, LLC, for the National No. DGE-1144085, No. AST-1207704, No. AST- Nuclear Security Administration of the USDOE (Contract 1207730, No. AST-1207752, No. MRI-1228509, No. 89233218CNA000001); the European Union’s No. OPP-1248097, No. AST-1310896, No. AST- Horizon 2020 research and innovation program under grant 1337663, No. AST-1440254, No. AST-1555365, agreement No. 730562 RadioNet; ALMA North America No. AST-1615796, No. AST-1715061, No. AST- Development Fund; the Academia Sinica; Chandra TM6- 1716327, No. AST-1716536, No. OISE-1743747, 17006X; Chandra award DD7-18089X. This work used the No. AST-1816420, No. AST-1903847, No. AST- Extreme Science and Engineering Discovery Environment 1935980, No. AST-2034306); the Natural Science (XSEDE), supported by NSF grant ACI-1548562, and Foundation of China (Grants No. 11573051, CyVerse, supported by NSF Grants No. DBI-0735191, No. 11633006, No. 11650110427, No. 10625314, No. DBI-1265383, and No. DBI-1743442. XSEDE

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Stampede2 resource at TACC was allocated through TG- by the NSF Physics Frontier Center Grant No. PHY- AST170024 and TG-AST080026N. XSEDE Jet-Stream 1125897 to the Kavli Institute of Cosmological Physics resource at PTI and TACC was allocated through at the University of Chicago, the Kavli Foundation and the AST170028. The simulations were performed in part on Gordon and Betty Moore Foundation grant GBMF 947. the SuperMUC cluster at the LRZ in Garching, on the The SPT hydrogen maser was provided on loan from the LOEWE cluster in CSC in Frankfurt, and on the HazelHen GLT, courtesy of ASIAA. The EHTC has received gen- cluster at the HLRS inStuttgart. This research was enabled erous donations of FPGA chips from Xilinx Inc., under the in part by support provided by Compute Ontario [99], Xilinx University Program. The EHTC has benefited from Calcul Quebec [100] and Compute Canada [101]. We thank technology shared under open-source license by the the staff at the participating observatories, correlation Collaboration for Astronomy Signal Processing and centers, and institutions for their enthusiastic support. Electronics Research (CASPER). The EHT project is This paper makes use of the following ALMA data: grateful to T4Science and Microsemi for their assistance ADS/JAO.ALMA\#2016.1.01154.V. ALMA is a partner- with Hydrogen Masers. This research has made use of ship of the European Southern Observatory (ESO; Europe, NASA’s Astrophysics Data System. We gratefully representing its member states), NSF, and National acknowledge the support provided by the extended staff Institutes of Natural Sciences of Japan, together with of the ALMA, both from the inception of the ALMA National Research Council (Canada), Ministry of Phasing Project through the observational campaigns of Science and Technology (MOST; Taiwan), Academia 2017 and 2018. We would like to thank A. Deller and W. Sinica Institute of Astronomy and Astro-physics(ASIAA; Brisken for EHT-specific support with the use of DiFX. We Taiwan), and Korea Astronomy and Space Science Institute acknowledge the significance that Maunakea, where the (KASI; Republic of Korea), in cooperation with the SMA and JCMT EHT stations are located, has for the Republic of Chile. The Joint ALMA Observatory is indigenous Hawaiian people. Facilities: EHT, ALMA, operated by ESO, Associated Universities, Inc. (AUI)/ APEX, IRAM:30 m, JCMT, LMT, SMA, ARO:SMT, NRAO, and the National Astronomical Observatory of SPT. Software: AIPS [102], ParselTongue [103], GNU Japan (NAOJ). The NRAO is a facility of the NSF operated Parallel [104], eht-imaging [105], Difmap [106], Numpy under cooperative agreement by AUI. This paper has made [107], Scipy [108], Pandas [109], Astropy [110,111], use of the following APEX data: Project ID T-091.F-0006- Jupyter [112], Matplotlib [113], THEMIS [114], DMC 2013. APEX is a collaboration between the Max-Planck- [115], polsolve [116], GPCAL [117]. Institut für Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden). The SMA is a joint APPENDIX: DISTORTION PARAMETERS project between the SAO and ASIAA and is funded by the Since the boundary of the shadow region is a closed curve Smithsonian Institution and the Academia Sinica. The as discussed above, one can define various characteristic JCMT is operated by the East Asian Observatory on behalf features for a quantitative comparison [80,83]. Out of the of the NAOJ, ASIAA, and KASI, as well as the Ministry of many possible measures of distortion of this curve from a Finance of China, Chinese Academy of Sciences, and the perfect circle discussed in Ref. [83], we use here the simplest National Key R&D Program (No. 2017YFA0402700) of one which was originally introduced in Ref. [80], namely China. Additional funding support for the JCMT is pro- α − α vided by the Science and Technologies Facility Council δ ¼ l;c l ð Þ sh ; A1 (UK) and participating universities in the UK and Canada. rsh;c The LMT is a project operated by the Instituto Nacional de where r is the radius of the circumcircle passing through Astrofísica, Óptica y Electrónica (Mexico) and the sh;c the two points (since the images here are symmetric University of Massachusetts at Amherst (USA), with about the α-axis) with coordinates ðα ; 0Þ and ðα ; β Þ, which financial support from the Consejo Nacional de Ciencia r t t are the rightmost and topmost points of the shadow curve, y Tecnología and the National Science Foundation. The and is given as [80], IRAM 30-m telescope on Pico Veleta, Spain is operated by IRAM and supported by CNRS (Centre National de la ðα − α Þ2 þ β2 ¼ t r t ð Þ Recherche Scientifique, France), MPG (Max-Planck- rsh;c 2jα − α j ; A2 Gesellschaft, Germany) and IGN (Instituto Geográfico t r ðα 0Þ ðα 0Þ Nacional, Spain). The SMT is operated by the Arizona with l; and l;c; the leftmost points of the shadow Radio Observatory, a part of the Steward Observatory of the curve and of the circumcircle respectively (see Fig. 3 University of Arizona, with financial support of operations of [57]). δ from the State of Arizona and financial support for In Fig. 4 we display the distortion parameter sh for the instrumentation development from the NSF. The SPT is shadow curves of various rotating black holes, for an supported by the National Science Foundation through equatorial observer, as an additional simple comparable δ Grant No. PLR-1248097. Partial support is also provided characteristic. We note also that the deviation of sh from

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As a concluding remark we note that the EHT bounds on the size of the shadow of M87*, as discussed above and displayed in Eq. (14), do not impose straightforward bounds on its shape. In particular, we can see from ¯ Fig. 4 that the rotating Bardeen black hole with qm ¼ 0.25 for high spins can be more distorted from a circle than a Kerr black hole but still be compatible with the EHT measurement (see Fig. 2). On the other hand, even though we are able to exclude Sen black holes with large ¯ ¼ electromagnetic charges (see, e.g., the Sen curve for qm 1.25 in the right panel of Fig. 2) as viable models for M87*, its shadow is less distorted from a circle than that of an extremal Kerr black hole (see Fig. 4). In other words, the examples just made highlight the importance of using the appropriate bounds on a sufficiently robust quantity when

FIG. 4. Distortion parameter δsh for a number of stationary using the EHT measurement to test theories of gravity. black holes observed on the equatorial plane (i ¼ π=2) with Failing to do so may lead to incorrect bounds on the black- dimensionless spin a. Because for observers viewing the black hole properties. For instance, Ref. [54] is able to set bounds hole from inclinations increasingly close to the pole, the shadow on the parameter space of the uncharged, rotating Hayward boundary appears increasingly circular, the distortions reported black hole by imposing bounds on the maximum distortion can be taken as upper limits. of the shape of its shadow boundaries, albeit using a different measure for the distortion from a circle [see zero is insignificant for observer viewing angles that are Eq. (58) there], whereas we have shown that this is not close to the pole of the black hole, as anticipated (not possible, upon using the bounds 4.31M − 6.08M for the displayed here). size of their shadows (cf. right panel of Fig. 2).

[1] R. H. Dicke, Republication of: The theoretical significance [10] E. Barausse and T. P. Sotiriou, Black holes in Lorentz- of experimental relativity, Gen. Relativ. Gravit. 51,57 violating gravity theories, Classical Quantum Gravity 30, (2019). 244010 (2013). [2] C. M. Will, The confrontation between general relativity [11] E. Barausse, T. P. Sotiriou, and I. Vega, Slowly rotating and experiment, Living Rev. Relativity 9, 3 (2006). black holes in Einstein-æther theory, Phys. Rev. D 93, [3] T. E. Collett, L. J. Oldham, R. J. Smith, M. W. Auger, 044044 (2016). K. B. Westfall, D. Bacon, R. C. Nichol, K. L. Masters, [12] O. Ramos and E. Barausse, Constraints on Horavaˇ gravity K. Koyama, and R. van den Bosch, A precise extra- from binary black hole observations, Phys. Rev. D 99, galactic test of General Relativity, Science 360,1342 024034 (2019). (2018). [13] O. Sarbach, E. Barausse, and J. A. Preciado-López, Well- [4] G. ’t Hooft and M. Veltman, One-loop divergencies in the posed Cauchy formulation for Einstein-æther theory, theory of gravitation, Ann. Inst. Henri Poincare Sect. A 20, Classical Quantum Gravity 36, 165007 (2019). 69 (1974). [14] T. Damour and J. H. Taylor, Strong-field tests of relativistic [5] N. V. Krasnikov, Nonlocal gauge theories, Theor. Math. gravity and binary pulsars, Phys. Rev. D 45, 1840 (1992). Phys. 73, 1184 (1987). [15] N. Wex, Testing relativistic gravity with radio pulsars, [6] H. Lü, A. Perkins, C. N. Pope, and K. S. Stelle, Black arXiv:1402.5594. Holes in Higher Derivative Gravity, Phys. Rev. Lett. 114, [16] N. Wex and M. Kramer, Gravity tests with radio pulsars, 171601 (2015). Universe 6, 156 (2020). [7] J. Scherk and J. H. Schwarz, How to get masses from extra [17] R. Abuter et al. (GRAVITY Collaboration), Detection of dimensions, Nucl. Phys. B153, 61 (1979). the gravitational redshift in the orbit of the star S2 near the [8] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Galactic centre massive black hole, Astron. Astrophys. Theory (Cambridge University Press, Cambridge, England, 615, L15 (2018). 1988). [18] R. Abuter, A. Amorim, M. Bauböck, J. P. Berger, H. [9] E. Barausse, T. Jacobson, and T. P. Sotiriou, Black holes in Bonnet, W. Brandner, V. Cardoso, Y. Cl´enet, P. T. de Einstein-aether and Horava-Lifshitzˇ gravity, Phys. Rev. D Zeeuw, and J. Dexter, Detection of the Schwarzschild 83, 124043 (2011). precession in the orbit of the star S2 near the Galactic

104047-15 PRASHANT KOCHERLAKOTA et al. PHYS. REV. D 103, 104047 (2021)

centre massive black hole, Astron. Astrophys. 636,L5 [32] T. Johannsen, Regular black hole metric with three con- (2020). stants of motion, Phys. Rev. D 88, 044002 (2013). [19] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. [33] L. Rezzolla and A. Zhidenko, New parametrization for Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, spherically symmetric black holes in metric theories of R. X. Adhikari et al., Observation of Gravitational Waves gravity, Phys. Rev. D 90, 084009 (2014). from a Binary Black Hole Merger, Phys. Rev. Lett. 116, [34] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya, and Y. 061102 (2016). Mizuno, New method for shadow calculations: Applica- [20] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. tion to parametrized axisymmetric black holes, Phys. Rev. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, D 94, 084025 (2016). R. X. Adhikari et al., Astrophysical implications of the [35] R. Konoplya, L. Rezzolla, and A. Zhidenko, General binary black hole merger GW150914, Astrophys. J. Lett. parametrization of axisymmetric black holes in metric 818, L22 (2016). theories of gravity, Phys. Rev. D 93, 064015 (2016). [21] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, [36] P. Kocherlakota and L. Rezzolla, Accurate mapping of A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. spherically symmetric black holes in a parametrized (Event Horizon Telescope Collaboration), First M87 event framework, Phys. Rev. D 102, 064058 (2020). horizon telescope results. I. The shadow of the super- [37] C. Bambi and K. Freese, Apparent shape of super-spinning massive black hole, Astrophys. J. Lett. 875, L1 (2019). black holes, Phys. Rev. D 79, 043002 (2009). [22] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, [38] C. Bambi and N. Yoshida, Shape and position of the A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. shadow in the δ ¼ 2 Tomimatsu–Sato spacetime, Classical (Event Horizon Telescope Collaboration), First M87 event Quantum Gravity 27, 205006 (2010). horizon telescope results. II. Array and instrumentation, [39] L. Amarilla, E. F. Eiroa, and G. Giribet, Null geodesics and Astrophys. J. Lett. 875, L2 (2019). shadow of a rotating black hole in extended Chern-Simons [23] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, modified gravity, Phys. Rev. D 81, 124045 (2010). A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. [40] L. Amarilla and E. F. Eiroa, Shadow of a Kaluza-Klein (Event Horizon Telescope Collaboration), First M87 event rotating dilaton black hole, Phys. Rev. D 87, 044057 horizon telescope results. III. Data processing and cali- (2013). bration, Astrophys. J. Lett. 875, L3 (2019). [41] S.-W. Wei and Y.-X. Liu, Observing the shadow of [24] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, Einstein-Maxwell-Dilaton-Axion black hole, J. Cosmol. A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. Astropart. Phys. 11 (2013) 063. (Event Horizon Telescope Collaboration), First M87 event [42] P. G. Nedkova, V. K. Tinchev, and S. S. Yazadjiev, Shadow horizon telescope results. IV. Imaging the central super- of a rotating traversable wormhole, Phys. Rev. D 88, massive black hole, Astrophys. J. Lett. 875, L4 (2019). 124019 (2013). [25] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, [43] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. Shadow of five-dimensional rotating Myers-Perry black (Event Horizon Telescope Collaboration), First M87 event hole, Phys. Rev. D 90, 024073 (2014). horizon telescope results. V. Physical origin of the asym- [44] S.-W. Wei, P. Cheng, Y. Zhong, and X.-N. Zhou, Shadow metric ring, Astrophys. J. Lett. 875, L5 (2019). of noncommutative geometry inspired black hole, J. [26] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, Cosmol. Astropart. Phys. 08 (2015) 004. A.-K. Baczko, D. Ball, M. Baloković, J. Barrett et al. [45] M. Ghasemi-Nodehi, Z. Li, and C. Bambi, Shadows of (Event Horizon Telescope Collaboration), First M87 event CPR black holes and tests of the Kerr metric, Eur. Phys. J. horizon telescope results. VI. The shadow and mass of the C 75, 315 (2015). central black hole, Astrophys. J. Lett. 875, L6 (2019). [46] F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Horizon [27] K. Gebhardt, J. Adams, D. Richstone, T. R. Lauer, S. M. structure of rotating Einstein–Born–Infeld black holes and Faber, K. Gültekin, J. Murphy, and S. Tremaine, The black shadow, Eur. Phys. J. C 76, 273 (2016). hole mass iN M87 from GEMINI/NIFS adaptive optics [47] B. P. Singh and S. G. Ghosh, Shadow of Schwarzschild- observations, Astrophys. J. 729, 119 (2011). Tangherlini black holes, arXiv:1707.07125. [28] D. Psaltis, L. Medeiros, P. Christian, F. Ozel, and the EHT [48] M. Amir, B. P. Singh, and S. G. Ghosh, Shadows of Collaboration, Gravitational Test beyond the First Post- rotating five-dimensional charged EMCS black holes, Newtonian Order with the Shadow of the M87 Black Hole, Eur. Phys. J. C 78, 399 (2018). Phys. Rev. Lett. 125, 141104 (2020). [49] H. Olivares, Z. Younsi, C. M. Fromm, M. De Laurentis, O. [29] T. Johannsen and D. Psaltis, Metric for rapidly spinning Porth, Y. Mizuno, H. Falcke, M. Kramer, and L. Rezzolla, black holes suitable for strong-field tests of the no-hair How to tell an accreting boson star from a black hole, Mon. theorem, Phys. Rev. D 83, 124015 (2011). Not. R. Astron. Soc. 497, 521 (2020). [30] T. Johannsen, Systematic study of event horizons and [50] Y. Mizuno, Z. Younsi, C. M. Fromm, O. Porth, M. De pathologies of parametrically deformed Kerr spacetimes, Laurentis, H. Olivares, H. Falcke, M. Kramer, and L. Phys. Rev. D 87, 124017 (2013). Rezzolla, The current ability to test theories of gravity with [31] S. Vigeland, N. Yunes, and L. C. Stein, Bumpy black holes black hole shadows, Nat. Astron. 2, 585 (2018). in alternative theories of gravity, Phys. Rev. D 83, 104027 [51] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, (2011). Spontaneously Scalarized Kerr Black Holes in Extended

104047-16 CONSTRAINTS ON BLACK-HOLE CHARGES WITH THE 2017 … PHYS. REV. D 103, 104047 (2021)

Scalar-Tensor–Gauss-Bonnet Gravity, Phys. Rev. Lett. [73] E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. 123, 011101 (2019). Prakash, and R. Torrence, Metric of a rotating, charged mass, [52] A. Grenzebach, The Shadow of Black Holes (Springer J. Math. Phys. (N.Y.) 6, 918 (1965). International Publishing, New York, 2016). [74] A. Sen, Rotating charged black hole solution in heterotic [53] Z. Stuchlík and J. Schee, Shadow of the regular Bardeen string theory, Phys. Rev. Lett. 69, 1006 (1992). black holes and comparison of the motion of photons and [75] C. Bambi and L. Modesto, Rotating regular black holes, neutrinos, Eur. Phys. J. C 79, 44 (2019). Phys. Lett. B 721, 329 (2013). [54] R. Kumar, S. G. Ghosh, and A. Wang, Shadow cast and [76] E. T. Newman and A. I. Janis, Note on the Kerr spinning‐ deflection of light by charged rotating regular black holes, particle metric, J. Math. Phys. (N.Y.) 6, 915 (1965). Phys. Rev. D 100, 124024 (2019). [77] B. Carter, Global structure of the Kerr family of gravita- [55] R. Kumar, A. Kumar, and S. G. Ghosh, Testing rotating tional fields, Phys. Rev. 174, 1559 (1968). regular metrics as candidates for astrophysical black holes, [78] F. Astorga, J. F. Salazar, and T. Zannias, On the integra- Astrophys. J. 896, 89 (2020). bility of the geodesic flow on a Friedmann–Robertson– [56] K. Hioki and U. Miyamoto, Hidden symmetries, null Walker spacetime, Phys. Scr. 93, 085205 (2018). geodesics, and photon capture in the Sen black hole, Phys. [79] S. Yazadjiev, LETTER: Newman–Janis method and rotat- Rev. D 78, 044007 (2008). ing dilaton-axion black hole, Gen. Relativ. Gravit. 32, [57] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, 2345 (2000). Shadow of rotating regular black holes, Phys. Rev. D 93, [80] K. Hioki and K.-I. Maeda, Measurement of the Kerr spin 104004 (2016). parameter by observation of a compact object’s shadow, [58] P. O. Mazur and E. Mottola, Gravitational vacuum con- Phys. Rev. D 80, 024042 (2009). densate stars, Proc. Natl. Acad. Sci. U.S.A. 101, 9545 [81] R. Shaikh, Black hole shadow in a general rotating (2004). spacetime obtained through Newman-Janis algorithm, [59] C. B. M. H. Chirenti and L. Rezzolla, Ergoregion insta- Phys. Rev. D 100, 024028 (2019). bility in rotating gravastars, Phys. Rev. D 78, 084011 [82] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating (2008). black holes: Locally nonrotating frames, energy extraction, [60] R. Shaikh, P. Kocherlakota, R. Narayan, and P. S. Joshi, and scalar synchrotron radiation, Astrophys. J. 178, 347 Shadows of spherically symmetric black holes and naked (1972). singularities, Mon. Not. R. Astron. Soc. 482, 52 (2019). [83] A. A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov, A [61] D. Dey, R. Shaikh, and P. S. Joshi, Shadow of nulllike and coordinate-independent characterization of a black hole timelike naked singularities without photon spheres, Phys. shadow, Mon. Not. R. Astron. Soc. 454, 2423 (2015). Rev. D 103, 024015 (2021). [84] T. Johannsen and D. Psaltis, Testing the no-hair theorem [62] R. M. Wald, General Relativity (The University of Chicago with observations in the electromagnetic spectrum. II. Press, Chicago, 1984). Black hole images, Astrophys. J. 718, 446 (2010). [63] J. Bardeen, in Proceedings of GR5, Tbilisi, USSR [85] M. E. Rodrigues and M. V. d. S. Silva, Bardeen regular (Georgia, 1968), p. 174. black hole with an electric source, J. Cosmol. Astropart. [64] S. A. Hayward, Formation and Evaporation of Nonsingular Phys. 06 (2018) 025. Black Holes, Phys. Rev. Lett. 96, 031103 (2006). [86] S. L. Liebling and C. Palenzuela, Dynamical boson stars, [65] V. P. Frolov, Notes on nonsingular models of black holes, Living Rev. Relativity 15, 6 (2012). Phys. Rev. D 94, 104056 (2016). [87] Z. Meliani, P. Grandcl´ement, F. Casse, F. H. Vincent, O. [66] D. I. Kazakov and S. N. Solodukhin, On quantum defor- Straub, and F. Dauvergne, GR-AMRVAC code applica- mation of the Schwarzschild solution, Nucl. Phys. B429, tions: accretion onto compact objects, boson stars versus 153 (1994). black holes, Classical Quantum Gravity 33, 155010 [67] G. W. Gibbons and K.-I. Maeda, Black holes and mem- (2016). branes in higher-dimensional theories with dilaton fields, [88] E. W. Hirschmann, L. Lehner, S. L. Liebling, and C. Nucl. Phys. B298, 741 (1988). Palenzuela, Black hole dynamics in Einstein-Maxwell- [68] D. Garfinkle, G. T. Horowitz, and A. Strominger, Charged dilaton theory, Phys. Rev. D 97, 064032 (2018). black holes in string theory, Phys. Rev. D 43, 3140 (1991). [89] J. Magueijo, New varying speed of light theories, Rep. [69] author A. García, D. Galtsov, and O. Kechkin, Class of Prog. Phys. 66, 2025 (2003). Stationary Axisymmetric Solutions of the Einstein-Max- [90] D. L. Wiltshire, M. Visser, and S. M. Scott, The Kerr well-Dilaton-Axion Field Equations, Phys. Rev. Lett. 74, Spacetime: Rotating Black Holes in General Relativity 1276 (1995). (Cambridge University Press, Cambridge, England, 2009). [70] R. Kallosh, A. Linde, T. Ortín, A. Peet, and A. van [91] A. Held, R. Gold, and A. Eichhorn, Asymptotic safety Proeyen, Supersymmetry as a cosmic censor, Phys. Rev. casts its shadow, J. Cosmol. Astropart. Phys. 06 (2019) D 46, 5278 (1992). 029. [71] A. I. Janis, E. T. Newman, and J. Winicour, Reality of the [92] C. Bambi, Testing the Bardeen metric with the black hole Schwarzschild Singularity, Phys. Rev. Lett. 20, 878 candidate in Cygnus X-1, Phys. Lett. B 730, 59 (2014). (1968). [93] E. Barausse, N. Yunes, and K. Chamberlain, Theory- [72] R. P. Kerr, Gravitational Field of a Spinning Mass as an Agnostic Constraints on Black-Hole Dipole Radiation Example of Algebraically Special Metrics, Phys. Rev. Lett. with Multiband Gravitational-Wave Astrophysics, Phys. 11, 237 (1963). Rev. Lett. 116, 241104 (2016).

104047-17 PRASHANT KOCHERLAKOTA et al. PHYS. REV. D 103, 104047 (2021)

[94] R. Konoplya and A. Zhidenko, Detection of gravitational [106] M. Shepherd, https://ui.adsabs.harvard.edu/abs/2011ascl waves from black holes: Is there a window for alternative .soft03001S. theories? Phys. Lett. B 756, 350 (2016). [107] S. van der Walt, S. C. Colbert, and G. Varoquaux, The [95] F.-L. Juli´e, On the motion of hairy black holes in Einstein- NumPy array: A structure for efficient numerical compu- Maxwell-dilaton theories, J. Cosmol. Astropart. Phys. 01 tation, Comput. Sci. Eng. 13, 22 (2011). (2018) 026. [108] E. Jones et al., SciPy: Open Source Scientific Tools for [96] F.-L. Juli´e, Gravitational radiation from compact binary Python (2001), http://www.scipy.org/. systems in Einstein-Maxwell-dilaton theories, J. Cosmol. [109] W. McKinney, Proc. IX Python in Science Conf., edited by Astropart. Phys. 10 (2018) 033. S. van der Walt and J. Millman (2010). [97] H. M. Siahaan, Merger estimates for Kerr-Sen black holes, [110] T. P. Robitaille et al., Astropy: A community Python Phys. Rev. D 101, 064036 (2020). package for astronomy, Astron. Astrophys. 558, A33 [98] S. H. Völkel, E. Barausse, N. Franchini, and A. E. Broderick, (2013). EHT tests of the strong-field regime of General Relativity, [111] A. M. Price-Whelan (The Astropy Collaboration), The arXiv:2011.06812. astropy project: Building an open-science project and [99] http://computeontario.ca. status of the v2.0 core package, Astron. J. 156, 123 (2018). [100] http://www.calculquebec.ca. [112] T. Kluyver et al., Positioning and Power in Academic [101] http://www.computecanada.ca. Publishing: Players, Agents and Agendas, edited by F. [102] E. W. Greisen, AIPS, the VLA, and the VLBA, in In: Heck Loizides and B. Schmidt (IOS Press, 2016). A. (eds) Information Handling in Astronomy - Historical [113] J. D. Hunter, Matplotlib: A 2D graphics environment, Vistas, edited by A. Heck, Astrophysics and Space Science Comput. Sci. Eng. 9, 90 (2007). Library Vol. 285 (Springer, Dordrecht, 2003), https://doi [114] A. E. Broderick, THEMIS: A parameter estimation frame- .org/10.1007/0-306-48080-8_7. work for the event horizon telescope, Astrophys. J. 897, [103] M. Kettenis, H. J. van Langevelde, C. Reynolds, and B. 139 (2020). Cotton, ParselTongue: AIPS Talking Python, in Astro- [115] D. W. Pesce, A D-term modeling code (DMC) for nomical Data Analysis Software and Systems XV, edited by simultaneous calibration and full-stokes imaging of very C. Gabriel, C. Arviset, D. Ponz, and S. Enrique, Astro- long baseline interferometric data, Astron. J. 161, 178 nomical Society of the Pacific Conference Series Vol. 351 (2021). (2006), https://ui.adsabs.harvard.edu/abs/2006ASPC..351. [116] I. Martí-Vidal, A. Mus, M. Janssen, P. de Vicente, .497K. and J. González, Polarization calibration techniques for [104] O. Tange, login: The USENIX Magazine 36, 42 (2011). the new-generation VLBI, Astron. Astrophys. 646,A52 [105] A. A. Chael, M. D. Johnson, R. Narayan, S. S. Doeleman, (2021). J. F. C. Wardle, and K. L. Bouman, High-resolution linear [117] J. Park, D.-Y. Byun, K. Asada, and Y. Yun, GPCAL: A polarimetric imaging for the event horizon telescope, generalized calibration pipeline for instrumental polariza- Astrophys. J. 829, 11 (2016). tion in VLBI data, Astrophys. J. 906, 85 (2021).

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